Bob,
There's no real advantage of using GPS over EDM, in fact it is a ton more complicated, time consuming, and expensive. EDM is over with a push of a button, twice to double-check. 10 minutes for a 1000ft course.
To get GPS to within a cm of true location, or distance, a couple options exist.
Surveyors use multi-channel GPS, and interperet data on two frequencies from up to 12 satellites simultaneously.
Method 1 - Real-time-kinematic or RTK- A fixed base station is set up to log satellite data, and sends correction data to the rover. The rover also collects satellite data, and can make corrections in real time to establish the coordinates relative to the base as the points are located.
After the field word has been done, the data is taken back to the office for analysis, and only there will you get a distance.
Essentially the addition of a base station creates another triangle to check for errors that occur due to atmospheric conditions, iononosphere activity, and various positional errors.
Method two - Static - Another way to do it is to set up the base station over both points, one at a time, and collect data in 2-3 sessions each lasting 3 hours minimum. After about three days of field-work and another couple hours to process the data, you'll know what a couple people can do in about 30 minutes with a tape. If the points are a few miles apart, then GPS becomes a viable method.